On parity problems and the functional-link artificial neural network

This paper contains the proof of a theorem on the capability of functional-link artificial neural networks both to represent and to learn the n-dimensional parity problem. The result is obtained by an embedding of the problem into a space of dimension 2ⁿ — 1. It is shown that the Volterra expansion of the data in n-dimensions provides the necessary transformation. By computing the parity function, it is shown that a suitable set of neural network weights can be deduced. Finally, it is demonstrated that 2ⁿ — 1 is the minimum embedding dimension for the problem.The contribution of A Zuderell of the University of Innsbruck is acknowledged.